A Digital WaveguideFilter (DWF) can be defined as any digital
filter built by stringing together digital waveguides.C.7 In the case of a cascade
combination of unit-length waveguide sections, they are essentially
the same as microwave filters [372] and unit element
filters [136] (Appendix F). This section shows how
DWFs constructed as a cascade chain of waveguide sections, and
reflectively terminated, can be transformed via elementary
manipulations to well known ladder and lattice digital
filters, such as those used in speech modeling
[297,363].

DWFs may be derived conceptually by sampling the unidirectional
traveling waves in a system of ideal, lossless waveguides. Sampling
is across time and space. Thus, variables in a DWF structure are
equal exactly (at the sampling times and positions, to within
numerical precision) to variables propagating in the physical medium
in an interconnection of uniform 1D waveguides.

To transform the DWF of Fig.C.24 to a conventional ladder
digital filter structure, as used in speech modeling
[297,363], we need to (1) terminate on the right
with a pure reflection and (2) eliminate the delays along the top
signal path. We will do this in stages so as to point out a valuable
intermediate case.

For velocity waves, the signs are interchanged. Thus, a reflectively
terminated ladder digital waveguide corresponds to a final feedback
with gain at the end of the ladder. Such a termination will
be used below to derive conventional ladder/lattice filters below.

The delays preceding the two inputs to each scattering junction can be
``pushed'' into the junction and then ``pulled'' out to the outputs
and combine with the delays there. (This is easy to show using the
Kelly-Lochbaum scattering junction derived in §C.8.4.)
By performing this operation on every other section in the DWF chain,
the half-rate ladder waveguidefilter shown in Fig.C.25 is
obtained [432].

figure[htbp]

This ``half-rate'' ladder DWF is so-called because its sampling rate
can be cut in half due to each delay being two-samples long. It has
advantages worth considering, which will become clear after we have
derived conventional ladder digital filters below. Note that now
pairs of scattering junctions can be computed in parallel, and
an exact physical interpretation remains. That is, it can still be
used as a general purpose physical modeling building block in this
more efficient (half-rate) form.

Note that when the sampling rate is halved, the physical wave
variables (computed by summing two traveling-wave components) are at
the same time only every other spatial sample. In particular, the
physical transverseforces on the right side of scattering junctions
and in Fig.C.25 are

respectively. In the half-rate case, adjacent spatial samples are
separated in time by half a temporal sample . If physical
variables are needed only for even-numbered (or odd-numbered) spatial
samples, then there is no relative time skew, but more generally,
things like collision detection, such as for slap-bass string-models
(§9.1.6), can be affected. In summary, the half-rate ladder
waveguide filter has an alternating half-sample time skew from section
to section when used as a physical modeling building block.

Given a reflecting termination on the right, the half-rate DWF chain
of Fig.C.25 can be reduced further to the conventional
ladder/lattice filter structure shown in Fig.C.26.

figure[htbp]

To make a standard ladder/lattice filter, the sampling rate is cut in
half (i.e., replace by ), and the scattering junctions are
typically implemented in one-multiply form (§C.8.5) or
normalized form (§C.8.6), etc. Conventionally, if the
graph of the scattering junction is nonplanar, as it is for the
one-multiply junction, the filter is called a lattice filter;
it is called a ladder filter when the graph is planar, as it is
for normalized and Kelly-Lochbaum scattering junctions. For all-poletransfer functions, the Durbin
recursion can be used to compute the reflection coefficients
from the desired transfer-function denominator polynomial coefficients
[449]. To implement arbitrary transfer-function zeros, a
linear combination of delay-element outputs is formed using weights
that are called ``tap parameters'' [173,297].

To create Fig.C.26 from Fig.C.24, all delays along the top rail
are pushed to the right until they have all been worked around to the
bottom rail. In the end, each bottom-rail delay becomes seconds
instead of seconds. Such an operation is possible because of the
termination at the right by an infinite (or zero) wave impedance.
Note that there is a progressive one-sample time advance from section
to section. The time skews for the right-going (or left-going)
traveling waves can be determined simply by considering how many
missing (or extra) delays there are between that signal and the
unshifted signals at the far left.

Due to the reflecting termination, conventional lattice filters cannot
be extended to the right in any physically meaningful way. Also,
creating network topologies more complex than a simple linear cascade
(or acyclic tree) of waveguide sections is not immediately possible
because of the delay-free path along the top rail. In particular, the
output cannot be fed back to the input . Nevertheless,
as we have derived, there is an exact physical interpretation (with
time skew) for the conventional ladder/lattice digital filter.

In some applications (such as time-varying waveguide reverberation
[430]), it may be preferable to compensate for the power
modulation so that changes in the wave impedances of the waveguides do
not affect the power of the signals propagating within.

In [432,433], three methods are discussed for making signal
power invariant with respect to time-varying branch impedances:

The normalized waveguide scheme compensates for power
modulation by scaling the signals leaving the delays so as to give
them the same power coming out as they had going in. It requires two
additional scaling multipliers per waveguide junction.

The transformer-normalized waveguide approach changes the
wave impedance at the output of the delay back to what it was at the
time it entered the delay using a ``transformer'' (defined in
§C.16).

The transformer-normalized DWF junction is shown in Fig.C.27
[432]. As derived in §C.16, the transformer ``turns
ratio'' is given by

We can now modulate a single scattering junction, even in arbitrary
network topologies, by inserting a transformer immediately to the left
or right of the junction. Conceptually, the wave impedance is not
changed over the delay-line portion of the waveguide section; instead,
it is changed to the new time-varying value just before (or after) it
meets the junction. When velocity is the wave variable, the
coefficients and in Fig.C.27 are swapped
(or inverted).

So, as in the normalized waveguide case, for the price of two extra
multiplies per section, we can implement time-varying digital filters
which do not modulate stored signal energy. Moreover, transformers
enable the scattering junctions to be varied independently, without
having to propagate time-varying impedance ratios throughout the
waveguide network.

It can be shown [433] that cascade waveguide chains built using
transformer-normalized waveguides are equivalent to those
using normalized-wave junctions. Thus, the transformer-normalized DWF
in Fig.C.27 and the wave-normalized DWF in Fig.C.22 are
equivalent. One simple proof is to start with a transformer
(§C.16) and a Kelly-Lochbaum junction (§C.8.4),
move the transformer scale factors inside the junction, combine terms,
and arrive at Fig.C.22. One practical benefit of this
equivalence is that the normalized ladder filter (NLF) can be
implemented using only three multiplies and three additions instead of
the usual four multiplies and two additions.